Entropies & Information Theory

Transcription

1 Etropes & Iformato Theory LECTURE II Nlajaa Datta Uversty of Cambrdge,U.K. See lecture otes o:

2 quatum system States (of a physcal system): Hlbert space (fte-dmesoal) H (state space) 0, Tr 1 desty matrces More geerally: f a quatum system s pure states: E,..., H, wth probs p, d 1 k k 1 Spectral decomposto: p, p,..., pk p geeral j j ; 0, : egevalues egevectors 1 1 d probablty dstrbuto d 1

3 Quatum Operatos or Quatum Chaels Ay allowed physcal process that a quatum system ca udergo s descrbed by a : lear completely-postve, trace preservg (CPTP) map ( ) put Trace-preservg (TP): Postve: ( ) 0 Completely postve (CP): : D ( H ) D( H ) A ( d )( ) E AE :CPTP map Tr Tr 1 B = a allowed state of the composte system AE ( d )( ) 0 E AE output A system E evromet D ( H H ) B E

4 Geeralzed measuremets POVM: A quatum measuremet s descrbed by a POVM E E ; E 0, E I (fte set) If the system s a state before the measuremet, th Tr( E) The, probablty of gettg the outcome s: p

5 Purfcato Ay mxed state H A A A pure state AR H ; A HR A Tr R AR AR ; purfyg referece system

6 of a state : Vo Neuma etropy S( ) : Tr log log log 2 Spectral decomposto: S( ) : Tr log d ; 1 d log 1 H Shao etropy S( ) 0 f ad oly f s a pure state: S( ) a measure of the mxedess of the state

7 Other Etropes For a bpartte system a state : AB A B Jot etropy: S( ) Tr( log ) AB AB AB Codtoal etropy: S A B) : S( ) S( ) ( AB B Quatum mutual formato: Tr B A AB reduced state I( A: B) : S( ) S( ) S( ); A B AB

8 Quatum Relatve Etropy A fudametal quatty Quatum Mechacs & Quatum Iformato Theory s the Quatum Relatve Etropy, 0, Tr 1, 0: of w.r.t. D( ): Tr log Tr lg o well-defed f supp supp log log 2 It acts as a paret quatty for the vo Neuma etropy: S( ) : Tr lo g D( I) ( I)

9 It also acts as a paret quatty for other etropes: e.g. for a bpartte state : Codtoal etropy AB A B S( A B) : S( ) S( ) D( I ) AB B AB A B Mutual formato Tr B A AB I( A: B) : S( ) S( ) S( ) D( ) A B AB AB A B

10 Some Propertes of D( ) dstace symmetrc tragle equalty D( ) 0 0, states f & oly f.(1) Mootocty uder a quatum operato (CPTP map) D( ( ) ( )) D( ).(2) May propertes of other etropes ca be proved usg (1) & (2)

11 Propertes of quatum etropes S( ) 0; S( ) log d; where d dm H Subaddtvty: Cocavty: H( X) H( X, Y) XY, : S( ) S( ) S( ) AB A B S p ps ( ) SU ( U ) S( ) Ivarace uder utares: but S( ) S( ) classcal r.v.s Codtoal etropy AB S( A B) A ca be egatve! s possble!! Arak-Leb equalty: S( ) S( ) S( ) AB A B

12 Propertes of quatum etropes cotd. Strog subaddtvty: ABC trpartte state S( ) S( ) S( ) S( ) ABC B AB BC B C A Leb & Ruska 73 Cosequeces of strog subaddtvty: Codtog reduces etropy S( A BC) S( A B) Dscardg quatum systems ever creases mutual formato I( A: B) I( A: BC) Quatum operatos ever crease mutual formato I( A: B') I( A: B) ; AB' A BB' AB (d )

13 Operatoal sgfcace of the vo Neuma etropy = optmal rate of data compresso for a memoryless (..d.) quatum formato source

14 Quatum Data Compresso Quatum Ifo source sgals (pure states) wth probabltes 1, 2..., r p1, p2,..., pr sgals H j j The source characterzed by:, H r p 1 desty matrx Memoryless quatum formato source State of copes of the source: o correlato

15 Quatum data compresso Evaluated the asymptotc lmt umber of copes/uses of the source 1, 2..., m emts sgals wth probs. State : Ecodg: E ( ) ( ) ( ) p, p,..., p m ( ) ( ) ( ) 1 2 m ( ) ( ) ( ) p 1 : H Compresso-Decompresso Scheme ( ) ( ) ( ) ( ) sgal compressed state D H c ( ) ( ) j j geeral compressed Hlbert space Decodg: D : D ( H ) ( ) ( ) recovered sgal

16 Quatum Data Compresso copes of a quatum formato source ( ) ( ) sgal B( H ) wth prob. ( ) p E ( ) compressed state dm H c H c dm H D ( ) recovered state B( H ) Requre: esemble average fdelty 1 F as...( a) F p D E ( ) ( ) ( ) ( ) ( ) Optmal rate of data compresso: Data compresso lmt R log(dm H c ) : lm Mmum value of such that holds ( a)

17 state of copes of the source ( ) : Memoryless quatum formato source m ( ) ( ) ( ) p 1 sgal emtted wth prob. D ( H ),dmh d Spectral decompostos: d q j j j j1 Idetfcato of the label ; ( p ) ; ; D ( H ) d ( ) ( ) j j ( ) ( ) ( ) k k k k 1... ( ) k k k k 1 2 q q... q k ( ) k k k k 1 2 as a sequece of classcal dces k k1 k2 k (,,..., )

18 state of copes of the source ( ) : Memoryless quatum formato source m ( ) ( ) ( ) p 1 sgal emtted wth prob. D ( H ),dmh d Spectral decompostos: d q j j j j1 Idetfcato of the label ; ( p ) ; ; ( ) ( ) j j H ( ) ( ) ( ) k k k k... ( ) k k k k 1 2 q q... q ( ) k k k k k k 1 2 as a sequece of classcal dces k k1 k2 k (,,..., )

19 ( ) ( ) ( ) k k k k q q... q ( ) k k k k 0, 1 2 vo Neuma etropy egevalues Probablty: egevectors S( ) S( ) S( ) k ( ) k k k... k q q q - sum over all possble sequeces k ( k1, k2,..., k ) : 1, 2,.., d ; 1 2 ( k1, k2,..., k ) ( H({ qk}) ) ( H({ qk}) ) 2 pk ( ) 2, a sequece ( ) k k ( ) p( k) ( S( ) ) ( ( ) ) 2 S pk ( ) 2, T ( ) : k sequeces s typcal f: k typcal subspace d dmh ({ }) H q k ( T ) : typcal set

20 typcal subspace T Subspace spaed by those egevectors... ( ) k k k k 1 2 H ( ) for whch k ( ) T Let ( P ) : orthogoal projecto o to the typcal subspace Typcal Sequece Theorem 0, 0, Fx the ad large eough: PT ( ) k (1 )2 1 T ( H({ q }) ) ( ) 2 k ( H({ q }) ) Typcal Subspace Theorem ( ) Tr 1 P (1 )2 dmt ( S( ) ) ( ) ( ( ) ) 2 S

21 Idea behd the compresso scheme ( ) : sgal emtted wth prob. ( p ) ; ( ) ( ) j j P I P ( ) ( ) ( ) ( ) ( ) ( ) T ( ) keep ths part uchaged T map ths oto a fxed pure state ( ) ( ) 0 T Compresso scheme E ( ) ( ) ( ) ( ) 2 ( ) ( ) 2 ( ) ( ) ( ) 0 0 D ( T ) 2 2 ( ) ( ) ( ) 2 ( ) ( ) 2 ( ) ( ) ; ; ( ) P P I P Decompresso D scheme ( ) ( ) 0

22 ( ) 2 ( ) ( ) 2 ( ) ( ) ( ) 0 0 D ( T ) Esemble average fdelty F p 2 p 1 ( ) ( ) ( ) ( ) ( ) 2

23 Schumacher proved (1995): for a memoryless source, H Data compresso lmt = S( ): vo Neuma etropy of the source

24 Schumacher s Theorem : Quatum Data Compresso, H Suppose s a memoryless, quatum formato source vo Neuma etropy R Suppose : the there exsts a relable compresso scheme of rate S( ) R ; S( ): for the source. R S( ) If the ay compresso scheme of rate wll ot be relable. R Proof follows from the Typcal Subspace theorem

25 Schumacher s Theorem : Quatum Data Compresso R S( ) R Suppose : the there exsts a relable compresso scheme of rate for the source. Proof: Compressed H R Hlbert space c; dm H c 2 R S( ) Choose such that RS( ) Fx 0, 0, choose large eough such that: ( ) Tr P 1 ; dm ( ) T ( ( ) ) 2 S R 2 =dm H c T ( ) H c

26 Esemble average fdelty F p 2 p 1 ( ) ( ) ( ) ( ) ( ) 2 12 (by the Typcal Subspace Theorem) F 1 as

27 Schumacher s Theorem : Quatum Data Compresso, H Suppose s a memoryless, quatum formato source vo Neuma etropy R Suppose : the there exsts a relable compresso scheme of rate S( ) R ; S( ): for the source. R S( ) If the ay compresso scheme of rate wll ot be relable. R (See Cambrdge lecture otes)